A decommissioned satellite splits into two pieces at an altitude, H, of 2000 km.

Question

A decommissioned satellite splits into two pieces at an altitude, H, of 2000 km. One piece falls directly to earth, while the other splits off at an angle, beta, and falls to earth. A space junk collector finds the first piece and then drives a distance, d_1 of 1500 km due east and finds the second piece. Given that the radius of the earth, R, is approximately 6378 km, determine the angle, beta, at which the satellite broke apart. Use the following diagram (not to scale) to solve the problem and assume that the rotation of the earth during the time between the landing of the pieces is negligible. Assume that the 1500 km (d_1) follows the curvature of a circle.

Explanation / Answer

in 2pi rad angle, arc is (2 pi R)

so in 1500 km arc angle will

alpha = (2pi / 2pi R) x 1500 = 0.235 rad =13.475 deg

in lower isosceles traingle.

(d2/2 ) = R sin(13.475/2)

R = 6378 km

d2 = 1496.54 km

now in upper right angle traingle,

tan(beta) = d2 / H

beta = tan^-1 ( 1496.54 / 2000) = 36.81 degrees.

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